Theorem blssex 12790

Description: Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)

Assertion

Ref	Expression
blssex	|- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )

Distinct variable groups:   x, r, A   D, r, x   P, r, x   X, r, x

Proof of Theorem blssex

Step	Hyp	Ref	Expression
1		blss 12788	. . . . . . 7 |- ( ( D e. ( *Met ` X ) /\ x e. ran ( ball ` D ) /\ P e. x ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ x )
2		sstr 3136	. . . . . . . . 9 |- ( ( ( P ( ball ` D ) r ) C_ x /\ x C_ A ) -> ( P ( ball ` D ) r ) C_ A )
3	2	expcom 115	. . . . . . . 8 |- ( x C_ A -> ( ( P ( ball ` D ) r ) C_ x -> ( P ( ball ` D ) r ) C_ A ) )
4	3	reximdv 2558	. . . . . . 7 |- ( x C_ A -> ( E. r e. RR+ ( P ( ball ` D ) r ) C_ x -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
5	1, 4	syl5com 29	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ x e. ran ( ball ` D ) /\ P e. x ) -> ( x C_ A -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
6	5	3expa 1185	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ x e. ran ( ball ` D ) ) /\ P e. x ) -> ( x C_ A -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
7	6	expimpd 361	. . . 4 |- ( ( D e. ( *Met ` X ) /\ x e. ran ( ball ` D ) ) -> ( ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
8	7	adantlr 469	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ x e. ran ( ball ` D ) ) -> ( ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
9	8	rexlimdva 2574	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) -> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )
10		simpll 519	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> D e. ( *Met ` X ) )
11		simplr 520	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> P e. X )
12		rpxr 9550	. . . . . 6 |- ( r e. RR+ -> r e. RR* )
13	12	ad2antrl 482	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> r e. RR* )
14		blelrn 12780	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ r e. RR* ) -> ( P ( ball ` D ) r ) e. ran ( ball ` D ) )
15	10, 11, 13, 14	syl3anc 1220	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> ( P ( ball ` D ) r ) e. ran ( ball ` D ) )
16		simprl 521	. . . . 5 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> r e. RR+ )
17		blcntr 12776	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ P e. X /\ r e. RR+ ) -> P e. ( P ( ball ` D ) r ) )
18	10, 11, 16, 17	syl3anc 1220	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> P e. ( P ( ball ` D ) r ) )
19		simprr 522	. . . 4 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> ( P ( ball ` D ) r ) C_ A )
20		eleq2 2221	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( P e. x <-> P e. ( P ( ball ` D ) r ) ) )
21		sseq1 3151	. . . . . 6 |- ( x = ( P ( ball ` D ) r ) -> ( x C_ A <-> ( P ( ball ` D ) r ) C_ A ) )
22	20, 21	anbi12d 465	. . . . 5 |- ( x = ( P ( ball ` D ) r ) -> ( ( P e. x /\ x C_ A ) <-> ( P e. ( P ( ball ` D ) r ) /\ ( P ( ball ` D ) r ) C_ A ) ) )
23	22	rspcev 2816	. . . 4 |- ( ( ( P ( ball ` D ) r ) e. ran ( ball ` D ) /\ ( P e. ( P ( ball ` D ) r ) /\ ( P ( ball ` D ) r ) C_ A ) ) -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) )
24	15, 18, 19, 23	syl12anc 1218	. . 3 |- ( ( ( D e. ( *Met ` X ) /\ P e. X ) /\ ( r e. RR+ /\ ( P ( ball ` D ) r ) C_ A ) ) -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) )
25	24	rexlimdvaa 2575	. 2 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. r e. RR+ ( P ( ball ` D ) r ) C_ A -> E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) ) )
26	9, 25	impbid 128	1 |- ( ( D e. ( *Met ` X ) /\ P e. X ) -> ( E. x e. ran ( ball ` D ) ( P e. x /\ x C_ A ) <-> E. r e. RR+ ( P ( ball ` D ) r ) C_ A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1335   e. wcel 2128   E.wrex 2436   C_ wss 3102   ran crn 4584   ` cfv 5167  (class class class)co 5818   RR*cxr 7894   RR+crp 9542   *Metcxmet 12340   ballcbl 12342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-po 4255  df-iso 4256  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-fv 5175  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-map 6588  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-xneg 9661  df-xadd 9662  df-psmet 12347  df-xmet 12348  df-bl 12350

This theorem is referenced by:  blbas  12793  elmopn2  12809  mopni2  12843  metss  12854  tgioo  12906